computer network

Question

Suppose that 2 parties A and B wish to setup a common secret key (D-H key) between themselves using the Diffie - Hellman key exchange technique. They agree on 47 as the modulus and 3 as the primitive root . Party A chooses 8 and party B chooses 10 as their respective secrets. Their D-H key is ___________________

A) 3

B) 4

C) 5

D) 6

Comments

https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange#Cryptographic_explanation

Answer 1

Difee Helman Algorithm

Step 1- A and B agree to use a prime number p and base g'
step 2- A choose a secret integer 'a' and then send AA=g^a mod p.
step 3- B choose a secret integer 'b' and then send BB=g^b mod p.
step 4- A compute secret key ,S=BB^a mod p.
step 5- B compute secret key ,S=AA^b mod p.

Answer -4

 

Comments

thank you :)

Answer 2

Let a be the primitive root and n be the modulus. Party A secret be KA and let party B secret be KB.

Party A is aware of a,n and KA whereas Party B is aware of a,n and KB. Message sent by A to B will be Mab = a^KA mod n and message sent by party B to A will be Mba = a^{KB mod n}

Mab = 3⁸ mod 47 

Mba = 3¹⁰ mod 47

The DH key for party A will be (Mba)⁸ mod 47 which is equal to DH key for party B which is, (Mab)¹⁰ mod 47.

(Mba)⁸ mod 47 = (Mab)¹⁰ mod 47 = (3¹⁰ mod 47)⁸ mod 47 = (3⁸ mod 47 )¹⁰ mod 47 = 3⁸⁰ mod 47 = 4

Comments

3⁸⁰ mod 47 = 4, I found out this value by exhaustive method. Can someone show the correct procedure to find value of such huge numbers ?

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3⁸⁰ mod 47 = (3⁵ mod 47)¹⁶ = (8 mod 47 )¹⁶ = (8² mod 47 )⁸ = (17 mod 47 )⁸ = (17² mod 47 )⁴

= (7 mod 47 )⁴​​​​​​​ = (7⁴ mod 47 )¹ = 4

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Thank you :)